1. Indian Employee visits US on L1 Visa and stays for a period less than 90 days.
His company is a indian subsidiary. His salary is paid by the Indian company (not the US parent company). Does he have to pay any tax in US ?

2. If his company rewards him with company shares then does he need to declare these shares? Does he need to pay the tax on these shares? If yes, what's the procedure?

Income that residents of India receive for personal services performed in the United States as employees (dependent personal services) is exempt from U.S. income tax if the residents meet three requirements.
• They are present in the United States for no more than 183 days during the tax year.
• The income is paid by, or on behalf of, an employer who is not a resident of the United States.
• The income is not borne by a permanent establishment, fixed base, or trade or business the employer has in the United States.
India U.S. Tax Treaty

Can you explain the 3rd point in detail?
"The income is not borne by a permanent establishment, fixed base, or trade or business the employer has in the United States"

Employee's salary is paid by the Indian company (not the US parent company). But the Indian Company (The branch in India) is run by the US parent Company. Does this means that the salary is borne by the fixed base? (Salary Slip of the employee shows the India address.)

If there is a U.S. parent company, then, yes, there IS a fixed based from which the employee is being paid.

Bottom Line: His earnings ARE subject to U.S. income taxes regardless of the duration of his stay.